Waves and Modes. Part I. Standing Waves. A. Modes

Transcription

1 Part I. Standing Waves Waves and Modes Whenever a wave (sound, heat, light,...) is confined to a finite region of space (string, pipe, cavity,... ), something remarkable happens the space fills up with a spectrum of vibrating patterns called standing waves. Confining a wave quantizes the frequency. Standing waves explain the production of sound by musical instruments and the existence of stationary states (energy levels) in atoms and molecules. Standing waves are set up on a guitar string when plucked, on a violin string when bowed, and on a piano string when struck. They are set up in the air inside an organ pipe, a flute, or a saxophone. They are set up on the plastic membrane of a drumhead, the metal disk of a cymbal, and the metal bar of a xylophone. They are set up in the electron cloud of an atom and the quark cloud of a proton. Standing waves are produced when you ring a bell, drop a coin, blow across an empty soda bottle, sing in a shower stall, or splash water in a bathtub. Standing waves exist in your mouth cavity when you speak and in your ear canal when you hear. Electromagnetic standing waves fill a laser cavity and a microwave oven. Quantum-mechanical standing waves fill the space inside carbon nanotubes and quantum computers. A. Modes A mass on a spring has one natural frequency at which it freely oscillates up and down. A stretched string with fixed ends can oscillate up and down with a whole spectrum of frequencies and patterns of vibration. Mass on Spring String with Fixed Ends (mass m, spring constant k) (length L, tension F, mass density µ) f = 2π k m f = 2L F µ Fundamenta l f 2 = 2f 2 nd harmonic f 3 = 3f 3 rd harmonic f 4 = 4f 4 th harmonic

2 These special Modes of Vibration of a string are called STANDING WAVES or NORMAL MODES. The word standing wave comes from the fact that each normal mode has wave properties (wavelength λ, frequency f), but the wave pattern (sinusoidal shape) does not travel left or right through space it stands still. Each segment (λ/2 arc) in the wave pattern simply oscillates up and down. During its up-down motion, each segment sweeps out a loop. Node λ/2 λ/2 f Node f Node A standing wave is a system of fixed nodes (separated by λ/2) and vibrating loops (frequency f). In short, a standing wave is a flip-flopping sine curve. All points on the string oscillate at the same frequency but with different amplitudes. Points that do not move (zero amplitude of oscillation) are called nodes. Points where the amplitude is maximum are called antinodes. The mathematical equation of a standing wave is y(x,t) = sin(2πx/λ) cos(2πft). The shape term sin(2πx/λ) describes the sinusoidal shape of the wave pattern of wavelength λ. The flip-flop term cos(2πft) describes the up-down oscillatory motion of each wave segment at frequency f. Each mode is characterized by a different λ and f. B. Harmonics The simplest normal mode, where the string vibrates in one loop, is labeled n = and is called the fundamental mode or the first harmonic. The second mode (n = 2), where the string vibrates in two loops, is called the second harmonic. The n th harmonic consists of n vibrating loops. The set of all normal modes {n =, 2, 3, 4, 5,... } is the harmonic spectrum. The spectrum of natural frequencies is {f, f 2, f 3, f 4, f 5,... }. Note that the frequency f n of mode n is simply a whole-number multiple of the fundamental frequency: f n = nf. The mode with 3 loops vibrates three times as fast as the mode with loop. Harmonics are the basis of HARMONY in music. The sectional vibrations of a string as one whole, two halves, three thirds, and so on, are very special because these vibrations produce musical tones that sound the most pleasant when sounded together, i.e. they represent the most harmonious combination of sounds. This explains the origin of the word harmonic. Exercise: Sketch the 6 th harmonic of the string. If the frequency of the 5 th harmonic is 00 Hz, what is the frequency of the 6 th harmonic? If the length of the string is 3 m, what is the wavelength of the 6 th harmonic? 2

3 Part II. Creating a Mode Resonance In general, when you pluck a string, you excite an infinite number of harmonic modes. How do you excite only one of the modes? There are three different methods:. The Mathematician Method: Sine Curve Initial Condition Pull Pull If you pull each mass element of a stretched string away from equilibrium (flat string) so that the string forms the shape of a sine curve, and then let go, the whole string will vibrate in one normal mode pattern. If you start with any other initial shape one that is not sinusoidal then the motion of the string will be made up of different modes. Starting out with an exact sine-curve shape is not easy to do you need some kind of fancy contraption. Pull 2. The Musician Method: Touch and Pluck Guitar players and violin players do this all the time. They gently touch the string at one point (where you want the node to be) and pluck the string at another point (antinode) to make a loop. The oscillation (up and down motion) of the plucked loop will drive the rest of the string to form additional equal-size loops which oscillate up and down at the same frequency as the driving loop. Pluck Touch 3. The Physicist Method: Resonance If you gently shake (vibrate) the end of a string up and down, a wave will travel to the right (R), hit the fixed end, and reflect back to the left (L). If you shake at just the right resonance frequency one that matches one of the natural frequencies of the string then the two traveling waves (R and L) will combine to produce a standing wave of large amplitude: R + L = STANDING WAVE. R+L R L Resonance Since RESONANCE is one of the most important concepts in science, we will focus on this idea. Resonance phenomena are everywhere: tuning a radio, making music, shattering a glass with your voice, imaging the body with an MRI machine, shaking cherry trees, designing lasers, engineering bridges, skyscrapers, and machine parts, etc. Consider pushing a person in a swing. If the frequency of your hand (periodic driving force) matches the natural frequency of the swing, then the swing will oscillate with large amplitude. It is a matter of timing, not strength. A sequence of gentle pushes applied at just the right time in perfect rhythm with the swing will cause a dramatic increase in the amplitude of the swing. A small stimulus gets amplified into a LARGE response. Experiment: Shake a Slinky, Make a Mode Go into the hallway. Stretch the slinky (in the air, not on the floor) so that its end-to-end length is 4 to 5 m. It is okay if the slinky sags a little. Keep one end fixed. GENTLY shake (vibrate) the other end side-to-side (side-to-side is easier than up-and-down). DO NOT shake with a large force or large amplitude. Remember that resonance is all in the timing, not the force! Shake at 3

4 just the right frequency to produce the two-loop (n = 2) normal mode. When you have tuned into this n = 2 state, take special note: You are now RESONATING with the coil. You and the coil are one! Your shaking hand is perfectly in-sync with the coil s very own natural vibration. The frequency of your hand matches f 2 of the coil. Note how the distance D hand that your hand moves back and forth is much smaller than the distance D coil that the coil moves back and forth. The driven slinky system has the ability to amplify a small input (D hand ) into a large output (D coil ). This is the trademark of resonance: a weak driving force (hand moves a little) causes a powerful motion (coil moves a lot). Estimate the values of D hand and D coil by simply observing the motions of your hand and the coil when you are resonating with the coil. Compute the Amplification Factor for this driven coil system. D hand = cm D coil = cm Amplification Factor : D coil / D hand =. Musical Interlude When you pluck a guitar string, bow a violin string, blow into a flute, or hit a drumhead, the object (string, air, plastic) will vibrate with a whole spectrum of modes all at the same time! Fourier s Theorem says that the general motion of any vibrating system can be represented by a superposition of mode motions. This is a far-reaching principle in science and engineering. When you pluck the A-string of a guitar, the resulting musical tone consists of the fundamental tone ( loop 00 Hz) plus the overtones 200 Hz, 300 Hz, 400 Hz, etc). Thought Experiment: Hands-on proof of Plucked String = Mixture of Harmonic Modes Suppose you pluck the A-string of a guitar and then lightly touch this vibrating string at its exact midpoint. Draw a picture of the plucked string before and after the touch. Assume that the fundamental (-loop) and the octave (2-loop) are much louder than the higher harmonics. Before After What tone do you hear after the touch? 4

5 Part III. Discovering a Harmonic Spectrum A. Experiment: Measuring the Spectrum Use the slinky as before keep one end fixed, make the length 4 to 5 meters, shake the other end. Record the length: L = m. Shake at the right frequencies to create the first three harmonics n =, 2, 3. Note: the uncertainty in measuring the integer n is exactly equal to zero: n ± 0! Measure the period of each harmonic. Note that the period of the flip-flopping loops on the slinky is equal to the period of your shaking hand. Use a stopwatch to measure the time for ten flip flops and then divide by ten to get the period. Sketch the shape of each mode you observed. Record your measured values of the period and the frequency of each mode. Mode Number Mode Shape Period (s) Frequency (Hz) 2 3 Do your measured frequencies satisfy the harmonic law f n = nf? Explain. B. Theory: Calculating the Spectrum Derive the f n formula. So far, you have measured the spectrum of natural frequencies of the slinky system. You will now calculate this spectrum based on the theory of waves. In the space below, combine the Kinematic Relation v = λf with the Harmonic Relation n(λ/2) = L to find how the frequency f n of mode n depends on the three variables n, v, and L. f n = To calculate the numerical value of f n from your theoretical formula, you will need to know the values of L and v for your slinky system. 5

6 Record L. Length of stretched slinky on which you observed n=,2,3: L = m. Find v. The velocity of a wave on a string (or a slinky) does not depend on the shape of the wave. All small disturbances propagate along the string at the same speed. So the best way to measure the value of v is to make a wave which is easiest to observe. The simplest kind of wave is a single pulse. Stretch the coil as before so that it has the same length L. Keep both ends fixed. At a point near one end, pull the slinky sidewise (perpendicular to the length) and release. This will generate a transverse traveling wave. Observe this wave disturbance (pulse) as it travels down the slinky, reflects off the other end, and travels back to its point of origin. Use a stopwatch to measure the time it takes for the pulse to make four round trips, i.e. 4 down-and-back motions along the entire length of the coil. Divide by 4 to get the round-trip time t r. Calculate the velocity v of the wave on your coil using the fact that the wave pulse travels a distance 2L (down L + back L) during the round trip time. Round-Trip Time t r = s. Wave Velocity v = m/s. Calculate f. Calculate the harmonic frequencies f, f 2, f 3 using your theoretical formula derived above. Show your calculation. C. Compare Experimental and Theoretical Spectrum Summarize your measured (Part A) and calculated (Part B) values of the natural frequencies {f, f 2, f 3 } that characterize your slinky system. n f n (measured) f n (calculated) % Difference 2 3 6

7 Part IV. Precision Measurement of Proper Tones and Wave Speed The system you will study consists of a string of finite length under a fixed tension. One end of the string is connected to an electric oscillator a high-tech shaker. Instead of you shaking the string by hand, the electric oscillator moves the string up and down at a frequency that you can control with high precision. The tension in the string is due to a precisely known mass hanging on the other end. Your research goal is to answer two basic questions:. What are the natural frequencies of this string system? 2. What is the speed of a wave on this system? Erwin Schrödinger, who won the Nobel Prize in Physics for the discovery of Quantum (Wave) Mechanics, referred to these natural frequencies as the proper tones of the system. A. Frequency Spectrum Hang 50 grams from the end of the string. Set up the standing waves n =, 2, 3, 4, 5 by driving the string with the electric oscillator using the following procedure. Procedure for Resonating with the String: Start with the frequency of the oscillator equal to.0 Hz. Set the amplitude of the oscillator equal to one-half its maximum value. Look at the end of the string attached to the oscillator. It is going up and down once every second just like your hand shaking the slinky! Now slowly increase the frequency until the first (n =) resonance is achieved where the string forms one big flip-flopping loop. If the oscillator makes a rattle sound, then decrease the amplitude. Continue to increase the frequency and successively tune into each of the overtones n = 2, 3, 4, 5. Record your measured values of f, f 2, f 3, f 4, f 5 : n f n (Hz) Do these natural frequencies (proper tones) of your string system form a harmonic spectrum? Explain. B. Wave Speed The wave speed on the string is too fast to measure with a stopwatch. Flick the string and try to observe the traveling pulse. Good Luck. Setting up standing waves is the gold-standard technique to measure the speed of any kind of wave from sound in air (300 m/s) to light in a vacuum (300,000,000 m/s). When you set up a standing wave, you are in effect freezing (taking a snapshot of) the speeding wave so you can easily observe the shape of the wave and measure the wavelength. 7

8 You will find the wave speed v using two different methods:. Space - Time Method: Measure λ and f for each mode n. Compute v = λf. You have already measured f for each n. To find λ for each n, measure the length L of your string (between it s two fixed endpoints) and then compute λ from the Fitting Condition: mode n consists of n half-waves stuffed inside L. The picture below shows the n = 3 fit. L λ/2 λ/2 λ/2 L = m. n λ (m) f (Hz) v = λf (m/s) average v = m/s. = mph. 2. Force - Mass Method: Measure F and µ of string. Compute v = (F/µ) /2. = m/s. The tension F in the string is equal to the weight of the hanging mass. To find the mass density µ of the string (µ mass per unit length), use the sample string on the back table. Do not remove the string attached to your oscillator and weight. Note: The sample string came from the same spool of string as the actual string and therefore has the same value of µ. The sample string may have a larger mass and a larger length than the actual string, but the mass-per-length ratio is the same. Tension: F = N. Mass density: µ mass / length = ( kg) / ( m) = kg/m. Wave Speed: v = (F/µ) /2 = m/s. Compare your two results for v [space-time λf and force-mass (F/µ) /2 ]: 8

9 Part V. Design Project: Create a 3-Loop 57 Hz The system consists of your stretched string, electric oscillator, and hanging weight. The application requires that you set up a 3-loop standing wave pattern on the string that vibrates at the rate of 57 cycles per second. The system parameter that you can change is the tension in the string. The Theory First work out the theory, then do the experiment. Recall the three essential elements of wave theory: () Kinematics: v = λf. (2) Dynamics: v = F/µ. (3) Harmonics: nλ/2 = L. The string parameters are length L, tension F, mass density µ, number of loops n. The wave properties are velocity v, frequency f, wavelength λ.. Carefully derive the formula that gives F as a function of the four parameters: n, L, f, µ. F = 2. Write the numerical values of the parameters: n =. L = m. f = Hz. µ = kg/m. 3. Substitute these values into your F formula to find the tension value: F = N. 4. Compute the value of the hanging mass: m = kg. The Experiment. Set the tension in the string by hanging the amount of mass as predicted by your theory. 2. Turn on the oscillator. 3. Vary the frequency of the oscillator so that the 3-loop mode is established on the string loud and clear, i.e. fine tune the frequency until the loop amplitude is maximum. 4. Record the value of this 3-loop frequency: f 3 = Hz. 5. Compare this experimental value of f 3 with the Design Specs value of 57 Hz. % difference between f 3 and 57 Hz is %. 9